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Article

Advancing Grey Modeling with a Novel Time-Varying Approach for Predicting Solar Energy Generation in the United States

by
Ke Zhou
1,
Ziji Zhao
2,
Lin Xia
3,* and
Jinghua Wu
3
1
Faculty of Management Engineering, Anhui Institute of Information Technology, Wuhu 241000, China
2
School of Electrical Engineering, Anhui Polytechnic University, Wuhu 241000, China
3
Faculty of Big Data Artificial Intelligence, Anhui Institute of Information Technology, Wuhu 241000, China
*
Author to whom correspondence should be addressed.
Sustainability 2024, 16(24), 11112; https://doi.org/10.3390/su162411112
Submission received: 12 November 2024 / Revised: 10 December 2024 / Accepted: 16 December 2024 / Published: 18 December 2024
Browse Figures
Figure 1
<p>The m-order time-varying function.</p> "> Figure 2
<p>Operating steps.</p> "> Figure 3
<p>U.S. solar energy generation and rate of increase from 2013 to 2023.</p> "> Figure 4
<p>Parameter selection process.</p> "> Figure 5
<p>The change trajectories of the fitted and predicted data obtained by eight methods.</p> "> Figure 5 Cont.
<p>The change trajectories of the fitted and predicted data obtained by eight methods.</p> "> Figure 6
<p>Comparison of MAPE of different methods.</p> "> Figure 7
<p>Comparison of forecasting APE of different methods.</p> "> Figure 8
<p>The future trend of solar energy generation in the U.S.</p> ">
Versions Notes

Abstract

:
This paper proposes a novel time-varying discrete grey model (TVDGM(1,1)) to precisely forecast solar energy generation in the United States. First, the model utilizes the anti-forgetting curve as the weight function for the accumulation of the original sequence, which effectively ensures the prioritization of new information within the model. Second, the time response function of the model is derived through mathematical induction, which effectively addresses the common jump errors encountered when transitioning from difference equations to differential equations in traditional grey models. Research shows that compared to seven other methods, this model achieves better predictive performance, with an error rate of only 2.95%. Finally, this method is applied to forecast future solar energy generation in the United States, and the results indicate an average annual growth rate of 23.67% from 2024 to 2030. This study advances grey modeling techniques using a novel time-varying approach while providing critical technical and data support for energy planning.

1. Introduction

1.1. Background

With the continuous advancement of science and technology, the demand for forecasting future developments across various fields is growing rapidly [1]. According to relevant statistical data, many industries today still heavily rely on traditional resources. This excessive dependence has led to numerous issues, accelerating systemic transformations on a global scale [2]. As the core of these systemic transformations, forecasting technologies play a critical role in optimizing resource allocation, reducing environmental impacts, and promoting sustainable development [3]. Therefore, in-depth research into future trend forecasting technologies is of great significance for achieving sustainable economic and social development.
However, many emerging fields face numerous challenges in forecasting due to limitations in data volume and the complexity of data characteristics. First, rapid policy changes result in insufficient availability of practical data. Second, since industry development is influenced by multiple factors, the associated data often exhibit significant nonlinear characteristics, increasing the difficulty of data analysis and modeling. Lastly, with the rapid development of the economy and technology, it has become especially important to prioritize the acquisition and processing of the latest information during the forecasting process to maintain the timeliness of predictions. Therefore, developing more advanced data modeling technologies to address these challenges has become a central task in forecasting research across various fields.

1.2. Literature Review

1.2.1. Summary of Forecasting Methods

Advancements in forecasting technology play a crucial role in providing technical support for energy management. At present, the main forecasting techniques in the energy sector encompass time series models, neural network models, combined forecasting models, and grey models. Time series analysis predicts future values by examining trends and patterns in historical data. Its advantages include simplicity, ease of use, and low computational requirements. This method has found extensive application in areas like electricity consumption (hydroelectric power [4], building electricity [5], industrial electricity [6]), wind speed [7], and reservoir usable capacity [8]. Neural network models make predictions by simulating the neural connections of the human brain. This method learns from large amounts of historical data to identify complex relationships between variables, enabling the prediction of future data. It has been widely applied in predicting building power load probability [9], shopping mall electricity consumption [10], CO2 emissions [11], office building energy consumption [12], chiller energy consumption [13,14], and subway chiller electricity consumption [15]. Combined forecasting models integrate multiple forecasting methods to enhance prediction accuracy and stability by leveraging the strengths of various approaches. These models have been broadly applied in predicting air pollution [16], power load [17], regional heating load [18], and wind energy generation [19].
Time series analysis is favored for its simplicity and low computational demand, but it is sensitive to the influence of outliers. Neural network methods offer powerful nonlinear mapping capabilities and adaptability, enabling them to handle complex nonlinear relationships and high-dimensional data. Although hybrid prediction models can optimize forecast accuracy and enhance the model’s stability and adaptability, they tend to be relatively complex. These methods have seen numerous successful applications in big data. However, their predictive capability is limited with complex limited samples.
The grey model is specifically designed to address situations where sample data are limited and conditions are uncertain. It uses known information to infer trends in unknown information, offering the advantages of low data requirements and high precision. Due to the limited and nonlinear characteristics of clean energy generation data, a grey model was chosen to make predictions in this field.

1.2.2. Grey Prediction Methods

As a fundamental part of grey system theory [20], the grey prediction model has been widely applied since Deng proposed grey theory [21], in areas such as economics [22], energy [23,24], transportation [25], and many others. Over the past 40 years, numerous scholars have expanded the grey prediction model from various perspectives, including seasonality [26], self-memory [27], compensation [28], and time delay [29].
Research on the Discrete Grey Model:
Xie et al. (2009) [30] proposed the discrete grey forecasting model. It is an effective tool for addressing uncertainty and prediction issues with small datasets. It simulates a system’s development trend by establishing differential equations and is continually refined in structure and algorithm to enhance prediction accuracy and adaptability. Traditional grey forecasting models can produce jump errors when transitioning from differential to micro-differential equations, but the discrete grey modeling method unifies parameter estimation and model solving, effectively avoiding these jump errors. Over the years, it has evolved into several models, including the approximated non-homogeneous exponential sequence forecasting model [31], the forecasting model with a linear time term [32], the forecasting model with a quadratic time term [33], and the forecasting model with a time lag function [34].
Research on Fractional-Order Grey Model:
Wu et al. (2013) [35] argued that data from different time points should be weighted differently, with an emphasis on prioritizing new information in the modeling process to thoroughly mine the grey information from system behavior sequences. Based on this principle, a fractional grey model is proposed. Ma et al. (2019) [36] constructed a discrete fractional grey multivariable model and used an improved grey wolf algorithm to find the optimal parameters. Pu et al. (2021) [37] developed a precise fractional grey Bernoulli model to forecast power usage. Wang et al. (2023) [38] proposed a structural adaptive fractional derivative grey model to predict energy consumption.

1.3. Summary

Traditional grey models (such as GM(1,1) and DGM(1,1), etc.) typically process the initial data using a first-order accumulation method. This approach treats all information equally, lacking the ability to extract the varying importance of the information. The fractional-order grey model takes into account the importance of information. Fractional grey models can ensure the prioritization of new information when the fractional accumulation operator r is between 0 and 1, but they cannot guarantee this priority when r is greater than 1. Therefore, a new grey model is needed to fully explore the importance of information and ensure the prioritization of new information.
To address these challenges, this paper develops a novel time-varying discrete grey model (TVDGM(1,1)). The main contributions include: (1) This article introduces a new weight accumulation function to replace the traditional first-order accumulation in the grey model, ensuring the prioritization of new information and enhancing the model’s adaptability to emerging trends. (2) The model is directly solved using mathematical induction, overcoming the jump error that occurs when traditional grey models transform from difference to differential forms. (3) The advantages of this model compared to other models have been validated, and it has successfully achieved accurate predictions of solar energy generation in the United States. The article provides important data support for energy planning in the United States. Overall, this paper proposes an effective method to improve the performance of the grey model, enhancing the model’s adaptability to future environments.

2. Methodology

2.1. The Traditional Discrete Grey Model

A discrete time series forecasting method employs the accumulated generating operation (AGO) from grey system theory to preprocess the original data sequence [30]. This model demonstrates the capability to accurately predict future data points under conditions of limited data and high uncertainty, serving as a simple yet effective tool for data analysis and prediction.
Definition 1.
Assume that X ( 0 ) = ( x 1 ( 0 ) , x 2 ( 0 ) , … , x n ( 0 ) ) is a non-negative original sequence. X ( 1 ) = ( x 1 ( 1 ) , x 2 ( 1 ) , … , x n ( 1 ) ) is the accumulative sequence, and x k ( 1 ) = ∑ i = 1 k x i ( 0 ) , k = 1,2 , … , n . The model is expressed as
x k + 1 ( 1 ) = β 1 x k ( 1 ) + β 2
where β 1 is the development coefficient, reflecting the trend of system changes. β 2 is the grey action quantity, reflecting external influences on the system.
Definition 2.
The estimate for parameter β ^ = ( β ^ 1 ,   β ^ 2 ) T of the model is calculated by the least square. β ^ = ( β ^ 1 ,   β ^ 2 ) T = ( B T B ) − 1 B T Y , and
B = x 1 1 , x 1 2 , … , x 1 n − 1 1 , 1 , … , 1 T ,
Y = ( x 1 2 , x ( 1 ) ( 3 ) , … , x ( 1 ) ( n ) ) T
Definition 3.
The time response function is obtained by
x ^ k + 1 ( 1 ) = β 1 k ( x 1 ( 0 ) − β 2 1 − β 1 ) + β 2 1 − β 1
Then,
x ^ k + 1 ( 0 ) = ( β 1 − 1 ) ( x 1 ( 0 ) − β 2 1 − β 1 ) β 1 k

2.2. Establishment of the Time-Varying Discrete Grey Model

Definition 4.
f i ( m ) is the m-order anti-forgetting function
f i ( m ) = e − m ( n + 1 − i ) , i = 1,2 , … , n
where m ( m ≥ 0 ) is defined as the stable variable of the anti-forgetting curve, which is applied to determine the rate of forgetting. i represents the location of the data in the original data vector. Figure 1 shows the visualization of the forgetting curve when m is fixed at 6 specific values.
Definition 5.
Assume that X ( 0 ) = ( x 1 ( 0 ) , x 2 ( 0 ) , … , x n ( 0 ) ) is a non-negative original, and x i ( 0 ) ≥ 0 , i = 1,2 , … , n . X ( m ) = ( x 1 ( m ) , x 2 ( m ) , … , x n ( m ) ) is the m-order time-varying cumulative sequence of X ( 0 ) and is defined as
x k ( m ) = ∑ i = 1 k f i ( m ) x i ( 0 ) , k = 1,2 , … , n . m > 0
Definition 6.
Assume that X ( 0 ) and X ( m ) are defined as shown in Definition 5, then the time-varying discrete grey model (TVDGM(1,1)) is described as follows:
x k + 1 ( m ) = β 1 x k ( m ) + β 2
where  β 1  and  β 2  are the linear structural parameters.
Theorem 1.
Assume that X ( 0 ) and X ( m ) are defined as shown in Definition 5. The estimate for parameter H ^ = [ β ^ 1 , β ^ 2 ] T of the model is calculated by the least square.
H ^ = [ β ^ 1 , β ^ 2 ] T = ( B T B ) − 1 B T Y
where
Y = x 2 ( m ) x 3 ( m ) ⋮ x n ( m ) , B = x 1 ( m ) 1 x 2 ( m ) 1 ⋮ ⋮ x n − 1 ( m ) 1
Theorem 2.
Assume that X ( 0 ) , X ( m ) and H ^ are defined as shown in Theorem 1, then the time response function of the TVDGM(1,1) is calculated as
x ^ k + 1 ( m ) = β ^ 1 k ( e − m n x 1 ( 0 ) − β ^ 2 1 − β ^ 1 ) + β ^ 2 1 − β ^ 1
Proof. 
Equation (5) is proved by the mathematical induction method.
When k = 1 , Equation (3) becomes
x ^ 2 ( m ) = β ^ 1 x ^ 1 ( m ) + β ^ 2 = β ^ 1 e − m n x 1 ( 0 ) + β ^ 2 = β ^ 1 ( e − m n x 1 ( 0 ) − β ^ 2 1 − β ^ 1 ) + β ^ 2 1 − β ^ 1
When k = 2 , Equation (3) becomes
x ^ 3 ( m ) = β ^ 1 x ^ 2 ( m ) + β ^ 2 = β ^ 1 β ^ 1 ( e − m n x 1 ( 0 ) − β ^ 2 1 − β ^ 1 ) + β ^ 2 1 − β ^ 1 + β ^ 2 = β ^ 1 2 ( e − m n x 1 ( 0 ) − β ^ 2 1 − β ^ 1 ) + β ^ 2 β ^ 1 1 − β ^ 1 + β ^ 2 = β ^ 1 2 ( e − m n x 1 ( 0 ) − β ^ 2 1 − β ^ 1 ) + β ^ 2 1 − β ^ 1
Assume that Equation (5) is true when k = d , d > 2 and
x ^ d + 1 ( m ) = β ^ 1 d ( e − m n x 1 ( 0 ) − β ^ 2 1 − β ^ 1 ) + β ^ 2 1 − β ^ 1
Then, when k = d + 1 , we have
x ^ d + 2 ( m ) = β ^ 1 x d + 1 ( m ) + β ^ 2 = β ^ 1 β ^ 1 d ( e − m n x 1 ( 0 ) − β ^ 2 1 − β ^ 1 ) + β ^ 2 1 − β ^ 1 + β ^ 2 = β ^ 1 d + 1 ( e − m n x 1 ( 0 ) − β ^ 2 1 − β ^ 1 ) + β ^ 2 β ^ 1 1 − β ^ 1 + β ^ 2 = β ^ 1 d + 1 ( e − m n x 1 ( 0 ) − β ^ 2 1 − β ^ 1 ) + β ^ 2 1 − β ^ 1
Hence, Equation (5) is correct when k = d + 1 . □
Theorem 1.
Assume that X ( 0 ) , X ( m ) , and H ^ are defined as shown in Theorem 1, then the fitted value of X ( 0 ) is calculated as
x ^ k + 1 ( 0 ) = ( β ^ 1 − 1 ) ( x 1 ( 0 ) − e m n β ^ 2 1 − β ^ 1 ) e − m k β ^ 1 k − 1
Proof. 
According to Equation (2), we can obtain x 1 ( m ) = e − m n x 1 ( 0 ) , and
x k + 1 ( m ) = ∑ i = 1 k + 1 e − m ( n + 1 − i ) x i ( 0 ) = ∑ i = 1 k e − m ( n + 1 − i ) x i ( 0 ) + e − m ( n + 1 − k − 1 ) x k + 1 ( 0 ) = x k ( m ) + e − m ( n − k ) x k + 1 ( 0 )
Therefore,
x ^ k + 1 ( 0 ) = e m ( n − k ) ( x ^ k + 1 ( m ) − x ^ k ( m ) ) = e m ( n − k ) ∗ [ β ^ 1 k ( e − m n x 1 ( 0 ) − β ^ 2 1 − β ^ 1 ) + β ^ 2 1 − β ^ 1 ] − [ β ^ 1 k − 1 ( e − m n x 1 ( 0 ) − β ^ 2 1 − β ^ 1 ) + β ^ 2 1 − β ^ 1 ] = e m ( n − k ) ∗ β ^ 1 k − 1 ( e − m n x 1 ( 0 ) − β ^ 2 1 − β ^ 1 ) ( β ^ 1 − 1 ) = ( β ^ 1 − 1 ) ( x 1 ( 0 ) − e m n β ^ 2 1 − β ^ 1 ) e − m k β ^ 1 k − 1
From the perspective of model structure, the TVDGM(1,1) has significant advantages. First, as time progresses, the values of the anti-forgetting function increase continuously. During the accumulation of initial values, using this function as a weight can effectively describe the varying weights of different information values. This approach helps ensure the prioritization of new information. Second, when the value of m is 0, this model can degenerate into the DGM(1,1), demonstrating its strong compatibility. However, when the data are relatively smooth, the model’s performance advantage is not as apparent compared to GM(1,1) and DGM(1,1). The TVDGM(1,1) shows more pronounced advantages when dealing with data that exhibit significant differences between new and old information. □

2.3. Optimization of Parameters

The model mainly includes three parameters ( β 1   , β 2 , m). ( β 1   , β 2 ) can be directly calculated through the least squares method. In this section, we primarily discuss the value of m. We use the whale optimization algorithm to select the optimal parameters. Compared to other methods, this algorithm is characterized by fast computation time and high accuracy [39]. We use MATLAB (2024a) for algorithm programming and computation.
Assuming X ( 0 ) are fitting sets, Equation (7) is a fitness function, and the constraints are designed as in Equation (8).
m = argmin ⁡ 1 n − 1 ∑ i = 2 n x ^ i ( 0 ) − x i ( 0 ) x i ( 0 )
f i ( m ) = e − m ( n + 1 − i ) , i = 1,2 , … , n x k ( m ) = ∑ i = 1 k f i ( m ) x i ( 0 ) , k = 1,2 , … , n . m ≻ 0 H ^ = [ β ^ 1 , β ^ 2 ] T = ( B T B ) − 1 B T Y x ^ k + 1 ( 0 ) = ( β ^ 1 − 1 ) ( x 1 ( 0 ) − e m n β ^ 2 1 − β ^ 1 ) e − m k β ^ 1 k − 1 1 n − 1 ∑ i = 2 n x ^ i ( 0 ) − x i ( 0 ) x i ( 0 )

2.4. Operating Steps

The operating steps of the TVDGM(1,1) method are described in Figure 2.
Step 1: Input original sequence X ( 0 ) . Step 2: Find the optimal parameter m by using the WOA. Step 3: Calculate parameters H ^ = [ β ^ 1 , β ^ 2 ] T = ( B T B ) − 1 B T Y by least squares method. Step 4: Calculate the time response function by Equation (5). Step 5: Obtain the prediction results from Equation (6).
Considering that the time complexity of the whale optimization algorithm has already been discussed in previous literature, this article focuses on the time complexity of the model itself. The calculation of the time complexity for the TVDGM(1,1) consists of three parts. First, the process of generating the m-order accumulated sequence has a time complexity of O ( n 2 ) . Second, solving the equations using the least squares method has a time complexity of O ( n 2 ) . Finally, the process of restoring the calculated values has a time complexity of O ( n ) . Therefore, the overall time complexity of the TVDGM(1,1) is O ( n 2 ) . From the perspective of the model itself, the time complexity of TVDGM(1,1) is the same as that of traditional models such as GM(1,1). Consequently, the additional time complexity compared to traditional models comes from the Whale Optimization Algorithm. Considering that grey models typically deal with small sample sizes, the time complexity of the optimization algorithm can essentially be ignored. Overall, the time complexity of the TVDGM(1,1) is relatively low.

3. Empirical Analysis and Discussion

Predicting solar energy generation in the United States is crucial for the country to develop energy planning and policies. This not only drives technological innovation and industrial upgrades but also helps achieve sustainable development goals and enhance international competitiveness and cooperation. This article introduces a new high-precision forecasting method and uses this method to predict solar power generation in the USA.

3.1. Data Collection and Analysis

Figure 3 displays the changes in solar power generation and its growth rate in the United States from 2013 to 2023. The data are sourced from BP (https://www.bp.com (accessed on 12 September 2024)). The overall trend of solar power generation in the United States is one of growth, with the growth rate showing nonlinear variations. This dataset validates that our model can assign different weights to historical data at different time points, enabling precise forecasting. We used data from 2013 to 2021 for model fitting, while data from 2022 to 2023 were used for testing. Based on the characteristics of U.S. solar energy generation data, the results demonstrate that this model has strong predictive capabilities.

3.2. Evaluation Index

The predicting effect is evaluated by APE and MAPE. APE measures the relative error between the forecasted and real values for a single sample point, used to evaluate the prediction precision of specific points. MAPE is the average of the absolute percentage errors of all sample points, reflecting the overall accuracy of the model’s predictions.
A P E i = x ^ i ( 0 ) − x i ( 0 ) x i ( 0 ) ,   M A P E = 1 n ∑ i = 1 n x ^ i ( 0 ) − x i ( 0 ) x i ( 0 )

3.3. Model Establishment

The operating steps are as follows:
Step 1. Input original sequence
X ( 0 ) = ( 16.04 , 29.22 , 39.43 , 55.42 , 78.06 , 94.31 , 107.97 , 132.04 , 166.08 )
Step 2. Find the optimal value of m through the WOA. Parameter selection follows the iterative process detailed in Figure 4. Ultimately, the optimal parameter determined is 0.5626.
Step 3. Calculate parameters H ^ = [ β ^ 1 , β ^ 2 ] T = ( B T B ) − 1 B T Y by least squares method. The results are as follows:
B = 0.10143730079172   1 0.42578259958195   1 1.19400726761738   1 3.08923790485781   1 7.77476193296337   1 17.7109875161555   1 37.6774384302927   1 80.5359776552163   1   Y = 0.42578259958195 1.19400726761738 3.08923790485781 7.77476193296337 17.7109875161555 37.6774384302927 80.5359776552163 175.156046706507
H ^ = [ β ^ 1 , β ^ 2 ] T = 2.1625 0.3011
Step 4. Calculate the time response function through Equation (5).
x ^ k + 1 ( 0.5626 ) = 2.162 5 k ( e − 0.5626 n x 1 ( 0 ) + 0.3011 1.1625 ) − 0.3011 1.1625
Step 5. Obtain the predicting results from Equation (6).
X ^ ( 0 ) = ( 16.04 ,   37.75 ,   46.51 ,   57.30 ,   70.60 ,   86.98 ,   107.17 ,   132.03 ,   162.67 ,   200.42 ,   246.92 )

3.4. Results and Discussion

The instance data completely confirm the superiority of the newly developed model in predicting solar energy generation in the United States. To more comprehensively demonstrate the model’s competitive advantage in predicting fluctuating time series, this paper compares the fitting and forecasting performance of eight different models using U.S. solar energy generation data. These models include the grey models, time series models, and the regression model, with specific results summarized in Table 1.
Table 2 describes the fitting and detection results of different methods. Although the TVDGM(1,1) is not the best in terms of fitting accuracy, its predictive capability during the testing phase significantly outperforms the other seven models. According to the statistics in Table 2, the TVDGM(1,1) demonstrates a marked advantage in forecasting performance.
Figure 5 illustrates the trajectory of changes in U.S. solar energy generation, comparing fitting and prediction data obtained using eight different models. The TVDGM(1,1) demonstrates the best alignment between its curve trend and the raw data. The other methods show varying degrees of deviation during the prediction phase, with performance significantly lower than that of the TVDGM(1,1). Notably, the ROGM(1,1), LR, TSQES, and TEP exhibit significant deviations in the prediction phase. This may be attributed to the simple structures of these models, which limit their ability to accurately predict U.S. solar energy generation. The TVDGM(1,1) effectively prioritizes new information, significantly enhancing its predictive accuracy.
Figure 6 presents a bar chart comparing the fitting and testing MAPE of different methods. Notably, the TVDGM(1,1) has a fitting error of 8.83%, slightly less than models like the ROGM(1,1), TSQES, and TEP. However, the testing error of the TVDGM(1,1) is only 2.95%, which is significantly superior to that of the other seven methods. This indicates that the model has a clear advantage in prediction. Among the evaluated models, methods like the ROGM(1,1), TSQES, and TEP, although having lower fitting errors, show testing errors exceeding 10%, indicating overfitting. Thus, these methods are unsuitable for predicting U.S. solar energy generation. The TMA and LR methods both have fitting errors exceeding 10%, similarly making them unsuitable. Although the GM(1,1) and DGM(1,1) have fitting and test errors below 10%, their predictive capabilities are noticeably weaker than those of the TVDGM(1,1). Overall, the TVDGM(1,1) method is better suited for predicting U.S. solar energy generation. This is mainly because the traditional grey model uses a first-order accumulation method, which overlooks the varying levels of importance among different pieces of information. In contrast, the model proposed in this paper emphasizes the significance of new information and adopts a novel approach to assign weights to different pieces of information.
Figure 7 presents the APE values at each point during the 2022–2023 testing phase for different methods. As shown, the TVDGM(1,1) exhibits lower and more stable APE values at each point during 2022–2023. Although the GM(1,1) had slightly lower test errors in 2022 compared to the TVDGM(1,1), its APE values were significantly higher at the time points in 2023. Furthermore, the APE values obtained with the TVDGM(1,1) were consistently lower than those from other methods. This indicates that the model we used demonstrates a high level of stability and long-term value in forecasting.
Overall, the TVDGM(1,1) achieved the best results during the predictive phase. These results suggest that incorporating time-varying weights aids in more effectively capturing the characteristics of U.S. solar energy generation data. The high predictive accuracy of this method holds significant practical application and reference value for forecasting U.S. solar energy generation. Thus, it is highly suggested as the optimal solution to tackle this forecasting issue.
We applied this method to forecast solar energy generation in the United States between 2024 and 2030. Table 3 shows the specific forecast results. In 2024, solar power generation in the United States is anticipated to reach 304.22 TWh, rising to 1064.1 TWh by 2030, with an average yearly increase of 23.67%. Figure 8 provides a visual representation of the data. As shown in the figure, solar energy generation in the United States exhibits a continuous upward trend during the forecast period. Examining Figure 8 reveals that the growth path of the forecast curve closely matches the actual data curve, which further confirms the accuracy and stability of the TVDGM(1,1) in data prediction.
Between 2024 and 2030, solar energy generation in the United States is projected to experience sustained growth, largely driven by several key factors. Firstly, enhanced federal and state policy support for renewable energy has been significant. This includes tax incentives and renewable energy quotas, which have created a conducive environment for the expansion of the solar market. Secondly, ongoing technological advancements have consistently reduced the costs associated with solar power generation and energy storage systems, thereby enhancing their economic viability. Furthermore, increased public awareness of environmental issues and climate change has heightened demand for clean energy solutions. Concurrently, corporations are increasingly integrating sustainability into their strategic frameworks, thereby accelerating investment in and deployment of solar projects. Collectively, these factors are expected to contribute to a steady increase in solar power generation.
Overall, this case study has three main characteristics. First, there are insufficient data available for prediction. Second, the data exhibit clear nonlinear characteristics. Third, the field is subject to significant policy and environmental changes, with new data having a greater impact on future trends. The TVDGM(1,1) performs exceptionally well in scenarios with small sample sizes and nonlinear data. This makes it applicable to other fields similar to solar energy generation forecasting in the United States, such as wind power generation, agricultural yield prediction, and environmental pollution trend analysis. These fields also face challenges such as limited data, significant nonlinear characteristics, and frequent changes in external conditions. However, the limitations of the model should also be noted, such as its potential performance issues when dealing with larger datasets or higher levels of noise.

4. Conclusions

This paper addresses the challenges posed by the complex nonlinear characteristics of data and the limited amount of available predictive data by proposing a novel time-variant discrete grey model (TVDGM(1,1)). This model advances grey modeling through an innovative time-variant approach and offers two notable advantages. First, it ensures the prioritization of new data by using the anti-forgetting curve as the weight accumulation function for the initial data, which directly contributes to the model’s superior accuracy, as evidenced by its error rate of only 2.95%—the lowest among the eight models tested. Second, the model’s solution process directly derives the response function using mathematical induction, effectively resolving the jump errors caused by transforming differences into differential equations in traditional grey models. The comparison of APE values and prediction trajectory performance across different models fully demonstrates that the model has good stability.
The model was applied to a case study of solar energy generation forecasting in the United States. The research results demonstrate that the proposed method exhibits outstanding predictive accuracy, significantly outperforming seven other models. This finding validates the innovations of the anti-forgetting curve and mathematical induction as key contributors to the model’s success. The method was subsequently applied to predict solar energy generation in the United States from 2024 to 2030, showing an average annual growth rate of 23.67%. These results hold significant implications for future energy planning in the United States. As a forward-looking solution, the TVDGM(1,1) has broad application potential and can be extended to other fields.
However, the study does have some limitations. Specifically, the TVDGM(1,1) is only suitable for univariate forecasting methods and overlooks other potential influencing factors. Future research will focus on extending the model from univariate to multivariate approaches, aiming to construct a multivariable time-variant grey model.

Author Contributions

Conceptualization, L.X. and J.W.; methodology, L.X. and K.Z.; investigation, Z.Z. and J.W.; writing—original draft preparation, K.Z. and L.X.; writing—review and editing, K.Z., L.X. and Z.Z.; supervision, L.X. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by Anhui Province young and middle-aged teachers training action project, grant number YQYB2023089; Research Projects of Higher Education Institutions in Anhui Province, grant number 2023AH052906; the Education Teaching Research Program of Anhui Provincial Department of Education, grant number 2022jyxm693; the University Natural Science Research Project of Anhui province, grant number KJ2020A0823; Wuhu City Science and Technology Research and Development Platform Project grant number No. 2023pt09; and the Open Research Fund of Anhui Province Key Laboratory of Machine Vision Inspection, grant number KLMVI-2024-HIT-17; Anhui Institute of Information Technology Research Center for Enterprise Innovation and Development, grant number 23kjcxpt003; Research Projects of Higher Education Institutions in Anhui Province, grant number 2024AH052572.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data presented in this study are available on request from the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

References

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Figure 1. The m-order time-varying function.
Figure 1. The m-order time-varying function.
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Figure 2. Operating steps.
Figure 2. Operating steps.
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Figure 3. U.S. solar energy generation and rate of increase from 2013 to 2023.
Figure 3. U.S. solar energy generation and rate of increase from 2013 to 2023.
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Figure 4. Parameter selection process.
Figure 4. Parameter selection process.
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Figure 5. The change trajectories of the fitted and predicted data obtained by eight methods.
Figure 5. The change trajectories of the fitted and predicted data obtained by eight methods.
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Figure 6. Comparison of MAPE of different methods.
Figure 6. Comparison of MAPE of different methods.
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Figure 7. Comparison of forecasting APE of different methods.
Figure 7. Comparison of forecasting APE of different methods.
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Figure 8. The future trend of solar energy generation in the U.S.
Figure 8. The future trend of solar energy generation in the U.S.
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Table 1. Abbreviations of different models.
Table 1. Abbreviations of different models.
IndexAbbreviationMeaning
1DCM(1,1)Discrete Grey Model [30]
2GM(1,1)Grey Model [20]
3ROGM(1,1)Random Oscillation Sequence Grey Model [40]
4TSQESTime Series Prediction with Quadratic Exponential Smoothing [41]
5TMATime Series Prediction with Trend Moving Average Method [41]
6TEPTime Series Prediction with Trend Extrapolation Prediction Method [42]
7LRLinear Regression Prediction [43]
8TVDGM(1,1)Time-Varying Discrete Grey Model (newly proposed method)
Table 2. Fitting and test results obtained by different methods.
Table 2. Fitting and test results obtained by different methods.
YearRaw DataTVDGM
(1,1)		GM
(1,1)		DGM
(1,1)		ROGM
(1,1)		LRTMATSQESTEP
201316.04----7.48 -16.04 14.20
201429.2237.75 37.01 37.27 29.22 25.56 -23.95 27.88
201539.4346.51 45.99 46.34 38.45 43.65 28.48 34.42 42.61
201655.4257.30 57.16 57.62 59.69 61.74 40.88 49.60 58.45
201778.0670.60 71.04 71.65 71.45 79.82 57.08 71.15 75.50
201894.3186.98 88.28 89.08 95.42 97.91 76.46 92.07 93.84
2019107.97107.17 109.72 110.77 110.14 116.00 93.66 110.73 113.58
2020132.04132.03 136.36 137.73 137.32 134.09 110.57 134.06 134.82
2021166.08162.67 169.47 171.25 155.50 152.17 134.53 165.74 157.68
Fitting MAPE (%) 8.83%8.60%9.10%4.30%12.38%21.19%6.31%5.08%
2022207.15200.42 210.62 212.93 186.44 188.35 192.64 181.08 182.28
2023240.53246.92 261.77 264.75 208.70 206.43 221.70 196.43 208.75
Test MAPE (%) 2.95%5.25%6.43%11.62%11.63%7.42%15.46%12.61%
Table 3. Forecast of solar energy generation in the U.S.
Table 3. Forecast of solar energy generation in the U.S.
Year2024202520262027202820292030
Solar energy generation (TWh)304.22374.81461.79568.94700.96863.611064.01
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Zhou, K.; Zhao, Z.; Xia, L.; Wu, J. Advancing Grey Modeling with a Novel Time-Varying Approach for Predicting Solar Energy Generation in the United States. Sustainability 2024, 16, 11112. https://doi.org/10.3390/su162411112

AMA Style

Zhou K, Zhao Z, Xia L, Wu J. Advancing Grey Modeling with a Novel Time-Varying Approach for Predicting Solar Energy Generation in the United States. Sustainability. 2024; 16(24):11112. https://doi.org/10.3390/su162411112

Chicago/Turabian Style

Zhou, Ke, Ziji Zhao, Lin Xia, and Jinghua Wu. 2024. "Advancing Grey Modeling with a Novel Time-Varying Approach for Predicting Solar Energy Generation in the United States" Sustainability 16, no. 24: 11112. https://doi.org/10.3390/su162411112

APA Style

Zhou, K., Zhao, Z., Xia, L., & Wu, J. (2024). Advancing Grey Modeling with a Novel Time-Varying Approach for Predicting Solar Energy Generation in the United States. Sustainability, 16(24), 11112. https://doi.org/10.3390/su162411112

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